# How you solve this? #lim_(n->oo)(|__x__|+|__3^2x__|+...+|__(2n-1)^2x__|)/n^3#

This can be understood as the realization of the Riemann-Stieltjes integral of

but

By signing up, you agree to our Terms of Service and Privacy Policy

Let us first find a closed formula for:

The first few terms are:

Write down the sequence of differences between consecutive terms:

Write down the sequence of differences of those differences:

Write down the sequence of differences of those differences:

Note also that:

So:

So:

By signing up, you agree to our Terms of Service and Privacy Policy

To solve the limit lim_(n->oo)(|**x**|+|**3^2x**|+...+|**(2n-1)^2x**|)/n^3, we can rewrite the expression as the sum of individual limits.

First, let's consider the limit of each term in the numerator.

For the term |**x**|, as n approaches infinity, x remains constant, so the limit is |x|.

For the term |**3^2x**|, the exponent 3^2x grows faster than n, so as n approaches infinity, this term becomes negligible and approaches 0.

Similarly, for the terms |**(2n-1)^2x**|, the exponents (2n-1)^2x also grow faster than n, so these terms also become negligible and approach 0 as n approaches infinity.

Therefore, the numerator simplifies to |x|.

The denominator, n^3, grows faster than all the terms in the numerator, so as n approaches infinity, the denominator dominates and the limit approaches 0.

In conclusion, the solution to the given limit is 0.

By signing up, you agree to our Terms of Service and Privacy Policy

To solve the limit ( \lim_{n \to \infty} \frac{|x| + |3^2x| + \ldots + |(2n - 1)^2x|}{n^3} ), we first recognize that as ( n ) approaches infinity, the expression inside the absolute values will keep growing larger since each term is squared. Therefore, we can simplify the absolute values as follows:

[ |(2k - 1)^2x| = (2k - 1)^2x ]

Where ( k ) represents the terms in the summation.

Now, let's rewrite the given expression:

[ \lim_{n \to \infty} \frac{|x| + |3^2x| + \ldots + |(2n - 1)^2x|}{n^3} ] [ = \lim_{n \to \infty} \frac{x + 3^2x + \ldots + (2n - 1)^2x}{n^3} ] [ = \lim_{n \to \infty} \frac{x(1^2 + 3^2 + \ldots + (2n - 1)^2)}{n^3} ]

Now, we recognize that the sum ( 1^2 + 3^2 + \ldots + (2n - 1)^2 ) is equivalent to the sum of the squares of the odd integers up to ( 2n - 1 ), which can be expressed as a known formula:

[ 1^2 + 3^2 + \ldots + (2n - 1)^2 = \frac{n(2n - 1)(2n + 1)}{3} ]

Substitute this into the expression:

[ \lim_{n \to \infty} \frac{x \cdot \frac{n(2n - 1)(2n + 1)}{3}}{n^3} ] [ = \lim_{n \to \infty} \frac{x(2n^3 + \ldots)}{3n^3} ] [ = \lim_{n \to \infty} \frac{x}{\frac{3}{2n} + \ldots} ]

As ( n ) approaches infinity, the terms with ( n ) in the denominator become negligible, leaving us with:

[ \lim_{n \to \infty} \frac{x}{\frac{3}{2n}} = \lim_{n \to \infty} \frac{2nx}{3} ]

Therefore, the limit evaluates to ( \frac{2x}{3} ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you evaluate the limit #(sqrt(x+1)-2)/(x-3)# as x approaches #3#?
- How do you find the limit of #(lnx)^2/sqrtx# as #x->oo#?
- How do you find the limit as x goes to 0 for the function #(3^x- 8^x)/ (x)#?
- What is the limit of #cos(1/x)# as x goes to infinity?
- How do you calculate this limit without using l’Hospital’s rule: lim √(x^2 +1) -1 / √(x^2+16) - 4 as x → 0 ?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7